Math 191 Wilkening
Spring 2023
Homework 3
due Sat, Feb. 11, 2:00 PM
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1. (2 points) (I.1.4, page 6, Strang.) Suppose A is the 3 by 3 matrix ones(3,3) of all ones. Find two independent vectors x and y that solve Ax = 0 and Ay = 0. Write that first equation Ax = 0 (with numbers) as a linear combination of the columns of A. Why don’t I ask for a third independent vector with Az = 0?
2. (3 points) (I.1.11, page 7, Strang.) Factor each of the following matrices into A = CR.
1 3 −2 A1=3 9 −6,
The matrix R will contain the numbers that multiply columns of C to recover columns of A.
0 A 3. (2 points) (I.1.18, page 7, Strang.) If A = CR, what are the CR factors of the matrix 0 A ?
(Here 0 stands for a matrix with the same number of rows as A and an arbitrary number of columns.)
4. (1 point) (I.2.3, page 13, Strang; slightly modified.) Suppose A ∈ Rm×n and B ∈ Rn×p. Denote B1
B2
the columns and rows of A and B by A = a1a2 · · · an and B = . , respectively.
(a) Give a “sum of rank one” formula for the matrix-matrix product AB.
(b) Use sigma notation to add the i,j entries of each matrix akBk found in part (a) to confirm that it agrees with the usual formula for (AB)ij.
5. (1 point) (I.2.6, page 13, Strang.) If A has columns a1, a2, a3 and B = I3×3 is the identity matrix, with rows B1, B2 and B3, what are the rank one matrices a1B1, a2B2 and a3B3? They should add to AI = A.
1 2 3 A2=4 5 6.
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6. (1 point) (I.3.3, page 20, Strang.) How is the nullspace of C related to the nullspaces of A and A
B, if C = B ?
7. (2 points) (I.3.4, page 20, Strang.) If the row space of A equals the column space of A, and also
N (A) = N (AT ), is A symmetric?
8. (1 point) (I.3.6, page 20, Strang.) Show that AT A has the same nullspace as A. Here is one approach: First, if Ax equals zero then AT Ax = ———. This proves N (A) ⊂ N (AT A). Second, if ATAx = 0, then xTATAx = ∥Ax∥2 = 0. Deduce N(ATA) = N(A).
9. (2 points) (I.3.10, page 20, Strang.) If N(A) is the zero vector, what vectors are in the nullspace ofB=[A A A]?
10. (3 points) (I.3.11, page 20, Strang.) For subspaces S and T of R10 with dimensions 2 and 7, what are all the possible dimensions of
(i) S ∩ T = {all vectors that are in both subspaces} (ii) S + T = {all sums s + t with s in S and t in T }
(iii) S⊥ = {all vectors in R10 that are perpendicular to every vector in S}.
11. (2 points) (I.4.3, page 27, Strang.) What lower triangular matrix E puts A into upper triangular form EA = U? Multiply by E−1 = L to factor A into LU:
2 1 0 A=0 4 2
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12. (3 points) (I.4.4, page 27, Strang.)
1 0 0 1 0 0 1 0 0
MultiplyA=a 1 0byE1 =−a 1 0andthenbyE2 =0 1 0. (Multyplyfromthe
b c 1 −b 0 1 0 −c 1 left, i.e., compute E1A and E2E1A. Do this before parts (a) and (b) below.)
(a) Multiply E2E1 to find the single matrix E that produces EA = I. (b) Multiply E−1E−1 to find the matrix A = L.
Note that the multipliers a, b, c are mixed up in E = L−1 but they are perfect in L.
13. (3 points) (I.4.6, page 27, Strang.) Which number c leads to a zero in the second pivot position? A row exchange is needed and A = LU will not be possible. Which c produces a zero in the third pivot position? Then a row exchange can’t help and elimination fails:
1 c 0 A=2 4 1
14. (2 points) (I.4.7, page 28, Strang.) Compute L and U for this symmetric matrix A:
a a a a A=a b b b a b c c
Find four conditions on a, b, c, d to get A = LU with four nonzero pivots.
15. (2 points) (I.4.8, page 28, Strang.) Tridiagonal matrices have zero entries except on the main diagonal and the two adjacent diagonals. Factor the following tridiagonal matrices into A = LU. Note that U = DLT , where D is diagonal with Dii = Uii, since these matrices are symmetric:
110 aa0 A1=1 2 1, A2=a a+b b .
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